Difficulty: Intermediate to Advanced | Prerequisites: Electrochemical cells (Part 1), redox half-reactions, basic thermodynamics (ΔG, equilibrium constants)
Big picture: This section ties electrochemistry to thermodynamics. Once you know which electrode is which, the next question is: how much voltage does the cell produce, and what does that tell you about spontaneity and equilibrium? The Nernst equation is the tool that extends standard-condition predictions to real-world concentrations. If the concepts of ΔG, K_eq, and Q are unfamiliar, review those before working through this material.
Every electrochemical cell has a measurable voltage (cell potential) determined by the difference in reduction potentials of its two half-cells. That voltage links directly to Gibbs free energy and the equilibrium constant, telling you whether a reaction is spontaneous and how far it will go. The Nernst equation lets you calculate the actual cell potential when concentrations are not at standard conditions.
Reduction potential (E°red)
The tendency of a species to gain electrons, measured in volts under standard conditions (1 M solutions, 1 atm gases, 25°C). Think of it as a scoreboard ranking: the more positive the value, the more eagerly that species accepts electrons.
Standard hydrogen electrode (SHE)
The reference electrode assigned a reduction potential of exactly 0 V. All other standard reduction potentials are measured relative to it. In simple terms, it is the zero point on the voltage scale.
Standard cell potential (E°cell)
The difference in reduction potentials between the cathode and anode half-cells under standard conditions. E°cell = E°red,cathode − E°red,anode. A positive value means the reaction is spontaneous as written.
Gibbs free energy (ΔG)
The thermodynamic quantity that determines spontaneity. For an electrochemical cell: ΔG° = −nFE°cell. A negative ΔG means the reaction proceeds spontaneously. Think of it as the energy "available" to do useful work.
Nernst equation
The equation that adjusts cell potential for non-standard concentrations: E = E° − (RT/nF) ln Q, or at 298 K: E = E° − (0.0592/n) log Q. In simple terms, this is how you calculate the real voltage when concentrations are not all 1 M.
Reaction quotient (Q)
The ratio of product concentrations to reactant concentrations at any given moment, using the same expression as the equilibrium constant but not restricted to equilibrium. Solids and pure liquids are omitted. As the reaction progresses, Q increases and E decreases.
Equilibrium constant (Keq)
The value of Q when the reaction has reached equilibrium. At equilibrium, E = 0 and ΔG = 0. Related to standard cell potential by E°cell = (0.0592/n) log Keq at 298 K.
Oxidising agent
A species that is easily reduced (gains electrons). The higher the reduction potential, the stronger the oxidising agent.
Reducing agent
A species that is easily oxidised (loses electrons). The lower (more negative) the reduction potential, the stronger the reducing agent.
Measured in volts, always under standard conditions (1 M, 1 atm, 25°C) for the standard value
The standard hydrogen electrode (SHE) is defined as E° = 0 V; all other values are relative to it
A more positive reduction potential means the species is more easily reduced (better oxidising agent)
A more negative reduction potential means the species is more easily oxidised (better reducing agent)
To find the potential of the reverse (oxidation) reaction, flip the sign of E°
In a galvanic cell, the electrode with the more positive E°red becomes the cathode
In an electrolytic cell, the electrode with the more positive E°red is forced to act as the anode by the external source
E°cell = E°red,cathode − E°red,anode
Some sources write this as E°cell = E°cathode + E°anode, where E°anode is the oxidation potential (sign already flipped); both give the same result
Positive E°cell: spontaneous reaction (galvanic cell)
Negative E°cell: non-spontaneous reaction (electrolytic cell)
Zero E°cell: system at equilibrium
ΔG° = −nFE°cell, where n = moles of electrons transferred and F = 96,485 C/mol
E°cell and ΔG° always have opposite signs:
Galvanic: positive E°cell, negative ΔG° (spontaneous)
Electrolytic: negative E°cell, positive ΔG° (non-spontaneous)
Equilibrium: E = 0, ΔG = 0
Must be at standard conditions (1 M, 1 atm, 25°C) to use the ° values directly
The work done by the cell depends on both the number of coulombs transferred (nF) and the potential (E)
Derived from ΔG = ΔG° + RT ln Q and ΔG = −nFE
Substituting: −nFE = −nFE° + RT ln Q
Dividing through by −nF: E = E° − (RT/nF) ln Q
At 298 K, converting ln to log: E = E° − (0.0592/n) log Q
As a reaction progresses, Q increases and E decreases
When Q = Keq, the system is at equilibrium and E = 0
Solids and pure liquids are omitted from the Q expression
At equilibrium, E = 0, so: 0 = E° − (0.0592/n) log Keq
Rearranging: E°cell = (0.0592/n) log Keq
If E°cell is positive at standard conditions, Keq > 1 (products favoured)
If E°cell is negative, Keq < 1 (reactants favoured)
If E°cell = 0, Keq = 1
For concentration cells with equimolar solutions, E°cell = 0 because both half-reactions are identical and there is no net ionic equation
Standard cell potential:
E°cell = E°red,cathode − E°red,anode
Gibbs free energy from cell potential:
ΔG° = −nFE°cell
Gibbs free energy from equilibrium constant:
ΔG° = −RT ln(Keq)
Gibbs free energy under non-standard conditions:
ΔG = ΔG° + RT ln(Q)
Nernst equation (general form):
E = E° − (RT / nF) ln(Q)
Nernst equation at 298 K:
E = E° − (0.0592 / n) log(Q)
Cell potential and equilibrium constant at 298 K:
E°cell = (0.0592 / n) log(Keq)
Key relationships at a glance:
Keq > 1 → E°cell positive → ΔG° negative → spontaneous
Keq < 1 → E°cell negative → ΔG° positive → non-spontaneous
Keq = 1 → E°cell = 0 → ΔG° = 0 → at equilibrium
The Nernst equation is the basis for how pH meters work: the voltage across a glass electrode changes predictably with hydrogen ion concentration, and the meter converts that voltage to a pH reading. Engineers use cell potential calculations to predict and prevent corrosion in pipelines, bridges, and ships, by selecting metals and coatings whose reduction potentials make unwanted oxidation thermodynamically unfavourable.
Students often mix up the sign convention for E°cell and ΔG°. They always have opposite signs. A positive E°cell means a negative ΔG°, and vice versa.
A frequent error is including solids or pure liquids in the reaction quotient Q. Only aqueous species and gases appear in Q.
Students sometimes think the Nernst equation only applies to galvanic cells. It applies to any electrochemical cell operating under non-standard conditions, including electrolytic and concentration cells.
Some students believe that E°cell changes when you multiply a half-reaction by a coefficient. It does not. Cell potential is an intensive property: doubling the half-reaction does not double E°. (However, n does change in ΔG° = −nFE°cell, so ΔG° does scale with stoichiometry.)
⚠️ Calculating E°cell from a table of standard reduction potentials is one of the most common exam questions. Identify the cathode (higher E°red) and the anode (lower E°red), then subtract.
⚠️ Expect a problem asking you to find ΔG° from E°cell, or vice versa, using ΔG° = −nFE°cell. Watch your signs and make sure n reflects the balanced equation.
⚠️ Nernst equation problems at 298 K are near-certain. You will be given non-standard concentrations and asked to calculate the actual cell potential. Remember to omit solids from Q.
⚠️ Linking E°cell to Keq is a common multi-step problem. Positive E°cell means Keq > 1 (products favoured). Be comfortable moving between E°, ΔG°, and Keq in any direction.
⚠️ Understanding that E° is intensive (does not change when you scale the equation) while ΔG° is extensive (does scale) is a subtle point that examiners test.
True or False: A positive E°cell corresponds to a positive ΔG°.
False. A positive E°cell corresponds to a negative ΔG° (spontaneous).
Fill in the blank: When Q = Keq, the cell potential E equals ______.
Zero.
True or False: Multiplying a half-reaction by 2 doubles its standard reduction potential.
False. E° is an intensive property and does not change with stoichiometric coefficients.
Fill in the blank: In the Nernst equation at 298 K, the factor in front of log Q is ______ / n.
0.0592.
True or False: If E°cell is negative, the equilibrium constant Keq is less than 1.
True.
Q: Given E°red(Cu²⁺/Cu) = +0.34 V and E°red(Zn²⁺/Zn) = −0.76 V, calculate E°cell for the Daniell cell and determine whether the reaction is spontaneous.
A: Cu is the cathode (higher E°red). E°cell = +0.34 − (−0.76) = +1.10 V. Since E°cell is positive, the reaction is spontaneous.
Q: For the Daniell cell above (n = 2), calculate ΔG° at 298 K.
A: ΔG° = −nFE°cell = −(2)(96,485)(1.10) = −212,267 J = −212.3 kJ. The negative value confirms spontaneity.
Q: Using the Nernst equation, calculate E for the Daniell cell at 298 K when [Zn²⁺] = 2.0 M and [Cu²⁺] = 0.010 M.
A: Q = [Zn²⁺]/[Cu²⁺] = 2.0/0.010 = 200. E = 1.10 − (0.0592/2) log(200) = 1.10 − (0.0296)(2.301) = 1.10 − 0.068 = 1.03 V. The cell still produces voltage, but less than under standard conditions because Q > 1.
Q: If E°cell for a reaction is +0.48 V and n = 3, calculate Keq at 298 K.
A: E°cell = (0.0592/n) log Keq, so log Keq = (0.48 × 3) / 0.0592 = 24.3. Keq = 10²⁴·³, an extremely large number, meaning products are overwhelmingly favoured at equilibrium.
Q: Why is E° not affected by multiplying a half-reaction by a coefficient, even though ΔG° is?
A: E° is a potential (energy per unit charge), which is an intensive property. Multiplying a half-reaction changes the number of electrons (n), which scales ΔG° = −nFE°, but the potential difference itself does not change.
This material builds directly on the electrochemical cells covered in the companion notes. The ΔG and Keq relationships tie back to your general thermodynamics and equilibrium chapters: if you can calculate ΔG from ΔH and ΔS, you can now also get it from E°cell, giving you three routes to the same answer. The Nernst equation reappears in biochemistry (membrane potentials, the Goldman equation) and analytical chemistry (potentiometric titrations, ion-selective electrodes).
cell potential, reduction potential, standard reduction potential, standard hydrogen electrode, SHE, electromotive force, emf, Gibbs free energy, ΔG, free energy and cell potential, Nernst equation, reaction quotient, Q expression, equilibrium constant, Keq and cell potential, spontaneity, intensive vs extensive properties, oxidising agent, reducing agent, 0.0592/n, Faraday's constant, voltage, half-cell potential, non-standard conditions, electrochemistry thermodynamics